Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

1  Introduction

In recent years, particularly in 2014, Khalil et al. [1] introduced a new definition of the fractional derivative called the conformable fractional derivative that extends the classical limit definition of the derivative of a function. The conformable fractional derivative has main advantages compared with other previous definitions. It can, for example, be used to solve the differential equations and systems exactly and numerically easily and efficiently, it satisfies the product rule and quotient rule, it has results similar to known theorems in classical calculus, and applications for conformable differential equations in a variety of fields have been extensively studied, see [2–10] and the references therein. On the other hand, in 2003, Khusainov et al. [11] represented the solutions of linear delay differential equations by constructing a new concept of a delayed exponential matrix function. In 2008, Khusainov et al. [12] adopted this approach to represent the solutions of an oscillating system with pure delay by establishing a delayed matrix sine and a delayed matrix cosine. This pioneering research yielded plenty of novel results on the representation of solutions, which are applied in the stability analysis and control problems of time-delay systems; see for example [13–28] and the references therein. Thereafter, in 2021, Xiao et al. [29] obtained the exact solutions of linear conformable fractional delay differential equations of order α∈(0,1] by constructing a new conformable delayed exponential matrix function.

However, to the best of our knowledge, no study exists dealing with the representation and stability of solutions of conformable fractional delay differential systems of order α∈(1,2].

Motivated by these papers, we consider the explicit formula of solutions of linear conformable fractional differential equations with pure delay

(D0αy)(x)=−By(x−τ)+f(x),for x≥0,τ>0,y(x)≡ψ(x),y′(x)≡ψ′(x)for - τ≤x≤0,(1)

by constructing new conformable delayed matrix functions. Moreover, the representation of solutions of Eq. (1) is used to obtain a finite time stability result on W=[0,L], L>0, where D0α is called the conformable fractional derivative of order α∈(1,2] with lower index zero, y(x)∈Rn, ψ∈C2([−τ,0],Rn), B∈Rn×n is a constant nonzero matrix and f∈C([0,∞),Rn) is a given function.

The paper is organized as follows: In Section 2, we present some basic definitions concerning conformable fractional derivative and finite time stability, and construct new conformable delayed matrix functions and derive their properties for use when we discuss the representation of solutions and finite time stability. In Section 3, by using the new conformable delayed matrix functions, we give the explicit formula of solutions of Eq. (1). In Section 4, as an application, we derive a finite time stability result using the representation of solutions. Finally, we give an example to illustrate the main results.

2  Preliminaries

Throughout the paper, we denote the vector norm and matrix norm, respectively, as ‖y‖=∑i=1n|yi| and ‖B‖=max1≤j≤n∑i=1n|bij|; yi and bij are the elements of the vector y and the matrix B, respectively. Denote C(W,Rn) the Banach space of vector-value continuous function from W→Rn endowed with the norm ‖y‖C=maxx∈W‖y(x)‖ for a norm ‖⋅‖ on Rn. We introduce a space C1(W,Rn)={y∈C(W,Rn):y′∈C(W,Rn)}. Furthermore, we see ‖ψ‖C=maxυ∈[−τ,0]‖ψ(υ)‖.

We recall some basic definitions of conformable fractional derivative, fractional exponential function, and finite time stability.

Definition 2.1. ([2, Definition 2.2]). Let f:[a,∞)→Rn be a differentiable function at x. Then the conformable fractional derivative for f of order α=(1,2] is given by

Daα(f)(x)=limε→0⁡f′(x+ε(x−a)2−α)−f′(x)ε,x>a,

if the limit exists.

Remark 2.1. As a consequence of Definition 2.1, we can show that

Daα(f)(x)=(x−a)2−αf′′(x),

where α=(1,2], and f is 2-differentiable at x>a.

Definition 2.2. ([2]). We define the fractional exponential function as follows:

Eα(λ,x−a)=exp⁡(λ.(x−a)αα)=∑k=0∞λk(x−a)αkαkk!,α>0,λ∈R.

Definition 2.3. ([30]). The system in Eq. (1) is finite time stable with respect to {0,W,τ,δ,β}, δ<β if and only if η<δ implies ‖y(x)‖<β for all x∈W, where η=max{‖ψ‖C,‖ψ′‖C,‖ψ′′‖C} and δ, β are real positive numbers.

Next, we construct new conformable delayed matrix functions that are the fundamental solution matrices of Eq. (1).

Definition 2.4. The conformable delayed matrix functions Hτ,α(Bxα) and Mτ,α(Bxα) are defined as

Hτ,α(Bxα):={Θ,−∞<x<−τ,I,−τ≤x<0,I−B1α(α−1)xα,0≤x<τ,⋮⋮I−B1α(α−1)xα+B212!α2(α−1)(2α−1)(x−τ)2α+⋯+(−1)mBm1m!αm∏i=1m(iα−1)(x−(m−1)τ)mα,(m - 1)τ≤x<mτ,(2)

Mτ,α(Bxα):={Θ,−∞<x<−τ,I(x+τ),−τ≤x<0,I(x+τ)−B1α(α+1)xα+10≤x<τ,⋮⋮I(x+τ)−B1α(α+1)xα+1+B212!α2(α+1)(2α+1)(x−τ)2α+1+⋯+(−1)mBm1m!αm∏i=1m(iα+1)(x−(m−1)τ)mα+1,(m - 1)τ≤x<mτ,(3)

respectively, where m=0,1,2,…, I is the n×n identity matrix and Θ is the n×n null matrix.

Lemma 2.1. The following rule is true:

D0αHh,α(Bxα)=−BHh,α(B(x−h)α).

Proof. First, when x∈(−∞,−τ), we obtain Hτ,α(Bxα)=Hτ,α(B(x−τ)α)=Θ, and we can see that Lemma 2.1 holds. Following that, set (m−1)τ≤x<mτ, m=0,1,2,…, we get

Hτ,α(Bxα)=I−B1α(α−1)xα+B212!α2(α−1)(2α−1)(x−τ)2α+⋯+(−1)mBm1m!αm∏i=1m(iα−1)(x−(m−1)τ)mα.

Applying Remark 2.1, we get

D0αHτ,α(Bxα)=D0αI−D0α[B1α(α−1)xα]+Dτα[B212!α2(α−1)(2α−1)(x−τ)2α]+⋯+D(m−1)τα[(−1)mBm1m!αm∏i=1m(iα−1)(x−(m−1)τ)mα]=Θ−B+B21α(α−1)(x−τ)α−B312!α2(α−1)(2α−1)(x−2τ)2α+⋯+(−1)mBm1(m−1)!αm−1∏i=1m−1(iα−1)(x−(m−1)τ)(m−1)α=−B[I−B1α(α−1)(x−τ)α+B212!α2(α−1)(2α−1)(x−2τ)2α+⋯+(−1)m−1Bm−11(m−1)!αm−1∏i=1m−1(iα−1)(x−(m−1)τ)(m−1)α]=−BHτ,α(B(x−τ)α).

This completes the proof.

In the same way that we proved Lemma 2.1, we can derive the next result.

Lemma 2.2. The following rule is true:

D0αMh,α(Bxα)=−BMh,α(B(x−h)α).

To conclude this section, we provide a norm estimation of the conformable delayed matrix functions, which is used while discussing finite time stability.

Lemma 2.3. For any x∈[(m−1)τ,mτ], m=0,1,2,…, we have

‖Hτ,α(Bxα)‖≤Eα(‖B‖α−1,x).

Proof. Taking the norm of Eq. (2), we get

‖Hτ,α(Bxα)‖≤1+‖B‖xαα(α−1)+‖B‖2(x−τ)2α2!α2(α−1)(2α−1)+⋯+‖B‖m(x−(m−1)τ)mαm!αm∏i=1m(iα−1)≤1+‖B‖xαα(α−1)+‖B‖2x2α2!α2(α−1)2+⋯+‖B‖mxmαm!αm(α−1)m≤∑k=0∞‖B‖kxαk(α−1)kαkk!=Eα(‖B‖α−1,x).

This completes the proof.

Lemma 2.4. For any x∈[(m−1)τ,mτ], m=0,1,2,…, we have

‖Mτ,α(Bxα)‖≤(x+τ)Eα(‖B‖α+1,x+τ).

Proof. Taking the norm of Eq. (3), we get

‖Mτ,α(Bxα)‖≤(x+τ)+‖B‖1α(α+1)xα+1+‖B‖212!α2(α+1)(2α+1)(x−τ)2α+1+⋯+‖B‖m1m!αm∏i=1m(iα+1)(x−(m−1)τ)mα+1≤(x+τ)+‖B‖(x+τ)α+1α(α+1)+‖B‖2(x+τ)2α+12α2(α+1)2+⋯+‖B‖m(x+τ)mα+1m!αm(α+1)m≤∑k=0∞‖B‖k(x+τ)kα+1k!αk(α+1)k=(x+τ)Eα(‖B‖α+1,x+τ).

This completes the proof.

3  Exact Solutions for Linear Conformable Fractional Delay Systems

In this section, we give the exact solutions of Eq. (1) via the conformable delayed matrix functions and the method of variation of constants. To do this, we consider the homogeneous system of linear conformable fractional delay differential equations

(D0αy)(x)=−By(x−τ),forx≥0,τ>0,y(x)≡ψ(x),y′(x)≡ψ′(x)for−τ≤x≤0,(4)

and the linear inhomogeneous conformable fractional delay system

(D0αy)(x)=−By(x−τ)+f(x),forx≥0,τ>0,y(x)≡Θ,y′(x)≡Θfor−τ≤x≤0.(5)

Theorem 3.1. The solution y(x) of Eq. (4) has the representation

y(x)={ψ(x),−τ≤x≤0,Hτ,α(Bxα)ψ(−τ)+Mτ,α(Bxα)ψ′(−τ)+∫−τ0Mτ,α(B(x−τ−υ)α)υα−2D0αψ(υ)dυ,x≥0.(6)

Proof. We seek for a solution of Eq. (4) in the form

y(x)=Hτ,α(Bxα)c1+Mτ,α(Bxα)c2+∫−τ0Mτ,α(B(x−τ−υ)α)υα−2D0αr(υ)dυ,(7)

or

y(x)=Hτ,α(Bxα)c1+Mτ,α(Bxα)c2+∫−τ0Mτ,α(B(x−τ−υ)α)r′′(υ)dυ,

where c1 and c2 are unknown constants vectors on Rn, and r(x) is an unkown twice continuously differentible vector function. From Lemmas 2.1 and 2.2, we deduce that Hτ,α(Bxα) and Mτ,α(Bxα) are solutions of Eq. (4). We notice that Eq. (6) is a solution of Eq. (4) due to the linearity of solutions for arbitrary c1, c2 and r(x). Now we find the constants c1 and c2, and the vector function r(x) so that the initial conditions y(x)≡ψ(x), y(x)≡ψ′(x) for −τ≤x≤0, are satisfied. That is, the following relations hold for −τ≤x≤0:

Hτ,α(Bxα)c1+Mτ,α(Bxα)c2+∫−τ0Mτ,α(B(x−τ−υ)α)r′′(υ)dυ=ψ(x),(8)

and

ddx{Hτ,α(Bxα)c1+Mτ,α(Bxα)c2+∫−τ0Mτ,α(B(x−τ−υ)α)r′′(υ)dυ}=ψ′(x).(9)

Consider Eq. (8). If −τ≤x<0, then

Hτ,α(Bxα)=I,Mτ,α(Bxα)=I(x + τ),

and

Mτ,α(B(x−τ−υ)α)={I(x−υ),υ∈[−τ,x],Θ,υ∈(x,0],

which implies that

c1+(x+τ)c2+∫−τx(x−υ)r′′(υ)dυ=ψ(x),(10)

and

∫−τx(x−υ)r′′(υ)dυ=−(x+τ)r′(−τ)+r(x)−r(−τ).(11)

Substituting Eq. (11) into Eq. (10), we get

(c1−r(−τ))+(c2−r′(−τ))(x+τ)+(r(x)−ψ(x))=Θ.(12)

Differentiating Eq. (12) with respect to x, we have

(c2−r′(−τ))+(r′(x)−ψ′(x))=Θ.(13)

As a result, we find that the equalities obtained Eqs. (12) and (13) are true if

c1=ψ(−τ),c2=ψ′(−τ),r(x)=ψ(x).(14)

Substituting Eq. (14) into Eq. (7), we obtain Eq. (6). This finishes the proof.

Theorem 3.2. The particular solution y0(x) of Eq. (5) has the representation

y0(x)=∫0xMτ,α(B(x−τ−υ)α)υα−2f(υ)dυ.(15)

Proof. We try to find a particular solution y0(x) of Eq. (5) in the form

y0(x)=∫0xMτ,α(B(x−τ−υ)α)ξ(υ)dυ,(16)

by applying the method of variation of constants, where ξ(υ), 0<υ≤x, is an unknown function. Taking the conformable derivative of Eq. (16), we get

D0αy0(x)=∫0xD0αMτ,α(B(x−τ−s)α)ξ(υ)dυ+x2−αξ(x)=−B∫0xMτ,α(B(x−2τ−υ)α)ξ(υ)dυ+x2−αξ(x).(17)

Substituting Eqs. (16) and (17) into Eq. (5), and noting that

∫x−τxMτ,α(B(x−2τ−υ)α)ξ(υ)dυ=Θ,

We have x2−αξ(x)=f(x). Substituting ξ(x)=xα−2f(x) into Eq. (16), we obtain Eq. (15). This completes the proof.

Corollary 3.1. The solution y(x) of Eq. (1) can be represented as

y(x)={ψ(x),−τ≤x≤0,Hτ,α(Bxα)ψ(−τ)+Mτ,α(Bxα)ψ′(−τ)+∫−τ0Mτ,α(B(x−τ−υ)α)υα−2D0αψ(υ)dυ+∫0xMτ,α(B(x−τ−υ)α)υα−2f(υ)dυ,x≥0.(18)

Remark 3.1. Let α=2 in Eq. (1). Then Corollary 3.1 coincides with Corollary 1 in [13].

Remark 3.2. Let α=2, B=B2 in Eq. (1) such that the matrix B is a nonsingular n×n matrix. Then

Hτ,2(B2x2)=cosτ(Bx),Mτ,2(B2x2)=B−1sinτ(Bx).

where cosτ(Bx) and sinτ(Bx) are called the delayed matrix of cosine and sine type, respectively, defined in [12]. Therefore, Corollary 3.1 coincides with Theorems 1 and 2 in [12].

4  Finite Time Stability of Linear Conformable Fractional Delay Systems

In this section, we establish some sufficient conditions for the finite time stability results of Eq. (1) by using a norm estimation of the conformable delayed matrix functions and the formula of general solutions of Eq. (1).

Theorem 4.1. The system Eq. (1) is finite time stable with respect to {0,W,τ,δ,β}, δ<β if

Eα(‖B‖α+1,L+τ)<β−δEα(‖B‖α−1,L)−‖f‖Cα(α−1)LαEα(‖B‖α+1,L)δ(L+τ)(τ+1).(19)

Proof. By using Definition 2.3, and Theorems 3.1 and 3.2, we have η<δ and

‖y(x)‖≤‖Hτ,α(Bxα)‖‖ψ(−τ)‖+‖Mτ,α(Bxα)‖‖ψ′(−τ)‖+‖∫−τ0Mτ,α(B(x−τ−υ)α)ψ′′(υ)dυ‖+‖∫0xMτ,α(B(x−τ−υ)α)υα−2f(υ)dυ‖≤δ‖Hτ,α(Bxα)‖+δ‖Mτ,α(Bxα)‖+δ∫−τ0‖Mτ,α(B(x−τ−υ)α)‖dυ+‖f‖C∫0x‖Mτ,α(B(x−τ−υ)α)‖υα−2dυ.(20)

Note that Mτ,α(Bxα)=Θ if x∈(−∞,−τ). For −τ≤υ≤0, we get

Mτ,α(B(x−τ−υ)α)={Mτ,α(B(x−τ−υ)α),υ∈[−τ,x],Θ,υ∈(x,0].

Thus

‖Mτ,α(B(x−τ−υ)α)‖={‖Mτ,α(B(x−τ−υ)α)‖,υ∈[−τ,x],0,υ∈(x,0].

Therefore, from Lemma 2.4, we have

‖Mτ,α(B(x−τ−υ)α)‖≤(x−υ)Eα(‖B‖α+1,x−υ)≤(x+τ)Eα(‖B‖α+1,x+τ),(21)

for −τ≤υ≤0, x∈W, and since Eα(‖B‖α+1,x−υ) is increasing function when x≥υ. From Eq. (21), we get

∫−τ0‖Mτ,α(B(x−τ−υ)α)‖dυ≤τ(x+τ)Eα(‖B‖α+1,x+τ).(22)

From Lemma 2.4, we have

∫0x‖Mτ,α(B(x−τ−υ)α)‖υα−2dυ≤∫0x(x−υ)Eα(‖B‖α+1,x−υ)υα−2dυ≤Eα(‖B‖α+1,x)∫0x(x−υ)υα−2dυ=xαα(α−1)Eα(‖B‖α+1,x).(23)

From Eqs. (20), (22) and (23), we get

‖y(x)‖≤δEα(‖B‖α−1,x)+δ(x+τ)Eα(‖B‖α+1,x+τ)+δτ(x+τ)Eα(‖B‖α+1,x+τ)+‖f‖Cα(α−1)xαEα(‖B‖α+1,x),(24)

for all x∈W. Combining Eq. (19) with Eq. (24), we obtain ‖y(x)‖<β for all x∈W. This completes the proof.

Corollary 4.1. Let α=2 in Eq. (1). Then the system

y′′(x)=−By(x−τ)+f(x),forx≥0,τ>0,y(x)≡ψ(x),y′(x)≡ψ′(x)for−τ≤x≤0,

is finite time stable with respect to {0,W,τ,δ,β}, δ<β if

E2(‖B‖3,L+τ)<β−δE2(‖B‖,L)−‖f‖C2L2E2(‖B‖3,L)δ(L+τ)(τ+1).

Remark 4.1. Let α=2, B=B2 in Eq. (1) such that the matrix B is a nonsingular n×n matrix. Then the representation of solution Eq. (18) coincides with the conclusion of Theorems 1 and 2 in [12], which leads to the same of the finite time stability results in [27].

5  An Example

Consider the conformable delay differential equations

(D01.8y)(x)=−By(x−0.5)+f(x),x∈[0,1],ψ(x)=(0.1x2,0.2x)T,ψ′(x)=(0.2x,0.2)T,ψ′′(x)=(0.2,0)T,−0.5≤x≤0,(25)

where

α=1.8,τ=0.5,B = (2002),f(x)=(x1/52x1/5).

From Theorems 3.1 and 3.2, for all 0≤x≤1, and through a basic calculation, we can obtain

y(x)=(0.025H0.5,1.8(2x1.8)−0.1H0.5,1.8(2x1.8))+(−0.1M0.5,1.8(2x1.8)0.2M0.5,1.8(2x1.8))+(0.2∫−0.50M0.5,1.8(2(x−0.5−υ)1.8)dυ0)+(∫0xM0.5,1.8(2(x−0.5−υ)1.8)dυ2∫0xM0.5,1.8(2(x−0.5−υ)1.8)dυ)=(y1(x)y2(x)),